The theory of semirings with applications in mathematics and theoretical computer science
The theory of semirings with applications in mathematics and theoretical computer science
A logical approach to discrete math
A logical approach to discrete math
ACM Transactions on Programming Languages and Systems (TOPLAS)
Maintaining knowledge about temporal intervals
Communications of the ACM
A Hardware Semantics Based on Temporal Intervals
Proceedings of the 10th Colloquium on Automata, Languages and Programming
A Duration Calculus with Infinite Intervals
FCT '95 Proceedings of the 10th International Symposium on Fundamentals of Computation Theory
Galois connections and fixed point calculus
Algebraic and coalgebraic methods in the mathematics of program construction
Complete Proof Systems for First Order Interval Temporal Logic
LICS '95 Proceedings of the 10th Annual IEEE Symposium on Logic in Computer Science
Duration Calculus: A Formal Approach to Real-Time Systems (Monographs in Theoretical Computer Science. an Eatcs Seris)
ACM Transactions on Computational Logic (TOCL)
IJCAI'85 Proceedings of the 9th international joint conference on Artificial intelligence - Volume 1
Towards an algebra of hybrid systems
RelMiCS'05 Proceedings of the 8th international conference on Relational Methods in Computer Science, Proceedings of the 3rd international conference on Applications of Kleene Algebra
Lazy semiring neighbours and some applications
RelMiCS'06/AKA'06 Proceedings of the 9th international conference on Relational Methods in Computer Science, and 4th international conference on Applications of Kleene Algebra
AMAST'06 Proceedings of the 11th international conference on Algebraic Methodology and Software Technology
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In 1996 Zhou and Hansen proposed a first-order interval logic called Neighbourhood Logic (NL) for specifying liveness and fairness of computing systems and defining notions of real analysis in terms of expanding modalities. After that, Roy and Zhou developed a sound and relatively complete Duration Calculus as an extension of NL. We present an embedding of NL into an idempotent semiring of intervals. This embedding allows us to extend NL from single intervals to sets of intervals as well as to extend the approach to arbitrary idempotent semirings. We show that most of the required properties follow directly from Galois connections, hence we get many properties for free. As one important result we obtain that some of the axioms which were postulated for NL can be dropped since they are theorems in our generalisation. Furthermore, we discuss other interval operations like Allen's 13 relations between intervals and their relationship to semiring neighbours. Then we present some possible interpretations for neighbours beyond interval settings. Here we discuss for example reachability in graphs and applications to hybrid systems. At the end of the paper we add finite and infinite iteration to NL and extend idempotent semirings to Kleene algebras and @w algebras. These extensions are useful for formulating properties of repetitive procedures like loops.